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Why Almost All k-Colorable Graphs Are Easy to Color

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Abstract

Coloring a k-colorable graph using k colors (k≥3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs lie in a single “cluster”, and agree on all but a small, though constant, portion of the vertices. We also describe a polynomial time algorithm that whp finds a proper k-coloring of such a random k-colorable graph, thus asserting that most such graphs are easy to color. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics.

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References

  1. Achlioptas, D., Friedgut, E.: A sharp threshold for k-colorability. Random Struct. Algorithms 14(1), 63–70 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Achlioptas, D., Moore, C.: Random k-sat: Two moments suffice to cross a sharp threshold. SIAM J. Comput. 36(3), 740–762 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Achlioptas, D., Ricci-Tersenghi, F.: On the solution-space geometry of random constraint satisfaction problems. In: Proc. 38th ACM Symp. on Theory of Computing

  4. Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput. 26(6), 1733–1748 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ben-Shimon, S., Krivelevich, M.: Random regular graphs of non-constant degree: edge distribution and applications. Manuscript (2006)

  6. Blum, A., Spencer, J.: Coloring random and semi-random k-colorable graphs. J. Algorithms 19(2), 204–234 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bollobás, B.: The chromatic number of random graphs. Combinatorica 8(1), 49–55 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  8. Böttcher, J.: Coloring sparse random k-colorable graphs in polynomial expected time. In: Proc. 30th International Symp. on Mathematical Foundations of Computer Science, pp. 156–167 (2005)

  9. Braunstein, A., Mézard, M., Weigt, M., Zecchina, R.: Constraint satisfaction by survey propagation. Comput. Complex. Stat. Phys. (2005)

  10. Braunstein, A., Mezard, M., Zecchina, R.: Survey propagation: an algorithm for satisfiability. Random Struct. Algorithms 27, 201–226 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Coja-Oghlan, A.: Coloring semirandom graphs optimally. In: Proc. 31st International Colloquium on Automata, Languages, and Programming, pp. 383–395 (2004)

  12. Coja-Oghlan, A., Krivelevich, M., Vilenchik, D.: Why almost all satisfiable k-cnf formulas are easy. In: 13th Conference on Analysis of Algorithms, DMTCS Proceedings, pp. 89–102 (2007)

  13. Daudé, H., Mézard, M., Mora, T., Zecchina, R.: Pairs of sat-assignments in random boolean formulæ. Theoret. Comput. Sci. 393(1–3), 260–279 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dyer, M.E., Frieze, A.M.: The solution of some random np-hard problems in polynomial expected time. J. Algorithms 10(4), 451–489 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. Syst. Sci. 57(2), 187–199 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Feige, U., Kilian, J.: Heuristics for semirandom graph problems. J. Comput. Syst. Sci. 63(4), 639–671 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Feige, U., Mossel, E., Vilenchik, D.: Complete convergence of message passing algorithms for some satisfiability problems. In: Random, pp. 339–350 (2006)

  18. Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs. Random Struct. Algorithms 27(2), 251–275 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Friedman, J., Kahn, J., Szemerédi, E.: On the second eigenvalue in random regular graphs. In: Proc. 21st ACM Symp. on Theory of Computing, pp. 587–598 (1989)

  20. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  21. Krivelevich, M., Vilenchik, D.: Semirandom models as benchmarks for coloring algorithms. In: 3rd Workshop on Analytic Algorithmics and Combinatorics, pp. 211–221 (2006)

  22. Kučera, L.: Expected behavior of graph coloring algorithms. In: Proc. Fundamentals of Computation Theory. Lecture Notes in Comput. Sci., vol. 56, pp. 447–451. Springer, Berlin (1977)

    Google Scholar 

  23. Łuczak, T.: The chromatic number of random graphs. Combinatorica 11(1), 45–54 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mulet, R., Pagnani, A., Weigt, M., Zecchina, R.: Coloring random graphs. Phys. Rev. Lett. 89(26), 268701 (2002)

    Article  MathSciNet  Google Scholar 

  25. Prömel, H., Steger, A.: Random l-colorable graphs. Random Struct. Algorithms 6, 21–37 (1995)

    Article  MATH  Google Scholar 

  26. Turner, J.S.: Almost all k-colorable graphs are easy to color. J. Algorithms 9(1), 63–82 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Informatics, University of Edinburgh, Edinburgh, UK

    Amin Coja-Oghlan

  2. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, 69978, Israel

    Michael Krivelevich

  3. Department of Mathematics, UCLA, Los Angeles, CA, 90095, USA

    Dan Vilenchik

Authors
  1. Amin Coja-Oghlan
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  2. Michael Krivelevich
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  3. Dan Vilenchik
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Corresponding author

Correspondence to Dan Vilenchik.

Additional information

Research of M. Krivelevich was supported in part by USA-Israel BSF Grant 2002-133, and by grant 526/05 from the Israel Science Foundation.

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Coja-Oghlan, A., Krivelevich, M. & Vilenchik, D. Why Almost All k-Colorable Graphs Are Easy to Color. Theory Comput Syst 46, 523–565 (2010). https://doi.org/10.1007/s00224-009-9231-5

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  • DOI: https://doi.org/10.1007/s00224-009-9231-5

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